## Computing the dimensions of representations in a reducible induced representation

This is a question on math.se that got no answers.

1) Is there a relatively general method of computing the dimensions of representations in a reducible induced representation?

An explicit specific example is: $G=Sp_4(\mathbb{F}_q)$, $J$ the antidiagonal $(1,1,-1,-1)$, $P$ the parabolic corresponding to $4=2+2$, $\tau$ a representation of $P$ that comes from a representation of $GL_2(\mathbb{F}_q)$, which will also be denoted $\tau_0$ (the matrix $[[g,*],[0,*]]$ acts as $\tau_0(g)$). Mackey's irreducibility criterion shows that $Ind_P^G \tau$ is reducible iff the central character $\omega_{\tau_0} =1$. How does one compute the dimensions of the (two) subrepresentations in this case? Note that this specific case is solved: "The characters of the finite symplectic group $Sp(4,q)$", B. Srinivasan, but the computations there are quite elaborate and not in the spirit of the question.

In the above example a short attempt to use Deligne-Lusztig theory seems to fail since the irreducible subrepresentations seem to be geometrically conjugate. Maybe the short attempt is ridiculous.

2) Is there a way to compute the dimensions, or even the characters themselves, using Deligne-Lusztig theory? (i.e. same question only restricted to reductive groups over finite fields)

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 Hindsight is always 20/20: Deligne-Lusztig theory works fine for the $Sp_4$ case, or, in the same spirit, for any parabolic induction of a cuspidal. From chapter 10 of Carter's book, mentioned in Jim's answer, we know the induction can decompose to up to 2 representations (in this specific case it also follows from [DL]). If it does, using Deligne-Lusztig, we know we have two pairs $(T,\theta)$, which are geometrically conjugate (as mentiond above). The maps $\rho_x$ and $\rho_x'$ near the end of [DL] now give the characters and dimensions of the two subrepresentations. – Dror Speiser Dec 9 2011 at 13:43

Small rank groups had been treated earlier in some detail, as noted in the question for $Sp_4$, but the general methods lead much farther than the ad hoc methods used in early papers.